ICTP/Psi-k/CECAM School on

Electron-Phonon Physics from First Principles

Trieste, 19-23 March 2018

Maximally-localized Wannier functionsGiovanni Pizzi1, Antimo Marrazzo1, Valerio Vitale2

1Theory and Simulation of Materials, EPFL (Switzerland) 2Cavendish Laboratory, Department of Physics, University of Cambridge (UK)

School on Electron-Phonon Physics from First Principles Trieste, March 20th, 2018

Lecture Tue.2

References• Marzari, N., and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)

• Souza, I., N. Marzari, and D. Vanderbilt, Phys. Rev. B 65, 035109 (2001)

• N. Marzari et al., Rev. Mod. Phys. 84, 1419–1475 (2012)

• R. M. Martin, Electronic Structure: Basic Theory and Practical Methods, Cambridge, 2004

• www.wannier.org

• First part of the slides: courtesy of Prof. Nicola Marzari.Can be found on the Wannier90 website: www.wannier.org under User Guide > NSF Summer School 2009 > N. Marzari Lecture Slides

PART I

Wannier functions

Crystal in real space:

Brillouin zone in reciprocal space:

0 –π/a π/a k

Bloch Theorem 

Courtesy of I. Souza / D. Vanderbilt 

Crystal in real space:

Brillouin zone in reciprocal space:

0 –π/a π/a k

Bloch Theorem 

Courtesy of I. Souza / D. Vanderbilt 

nk(r) = unk(r)eik·r

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defined on the wholeunit cell and periodic

over the cells

k(r) = uk(r)eik·r

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If there is only one band (n=1):

And we can define the Wannier functions:

|Ri =Z

BZ k(r)e

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- One WF per lattice vector R: N in total with Born-von Karman PBC with N total unit cells

- They are all identical, only shifted: if we have they are shifted by R2 - R1

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From Bloch Orbitals to Wannier Func:ons 

Multiband case, simplest thing to do:

Note: The shape of the WFs (in real space) will be different for every phase!

From Bloch Orbitals to Wannier Func:ons 

Multiband case, simplest thing to do:

More generally:

n=1

n=2

0 –/a /a

k

Unitary matrix Rotated Bloch function

Orthogonal and unitary transforma:ons 

Courtesy of I. Souza / D. Vanderbilt

Crystal in real space:

Brillouin zone in reciprocal space:

0 –π/a π/a k

Bloch Theorem 

Courtesy of I. Souza / D. Vanderbilt 

Generalized Wannier Func:ons for Composite Bands 

EachunitarymatrixchoosesadifferentsetofWFs.Wewouldliketochoosethe“best”,i.e.the“maximally-localized”

The Localiza:on Func:onal 

(Foster‐Boys) 

N. Marzari and D. Vanderbilt, Phys. Rev. B 56, 12847 (1997)

Decomposi:on of the Localiza:on

 Func:onal 

Centers of Wannier func:ons: 

Blount iden::es 

Decomposi:on of the Localiza:on

 Func:onal 

WF center

definition

Bloch theorem

How to compute?Blount identities

The Reciprocal Space Representa:on 

Blount iden::es 

Numerical approach: numerical derivatives on a uniform k grid in the BZ

The Reciprocal Space Representa:on 

Blount iden::es 

Therefore:

We can express the relevant quantities as a function of the Mmnmatrices (these willbe one of the main inputs to Wannier90)

The Reciprocal Space Representa:on 

Blount iden::es 

Numerical approach: numerical derivatives on a uniform k grid in the BZ

The Reciprocal Space Representa:on 

We can express the relevant quantities as a function of the Mmnmatrices (these willbe one of the main inputs to Wannier90)

To compute the maximal localization, we do not need to know the wavefunctions, but only the

overlaps Mmn matrices at neighbouring k-points

(after minimization, if we want to plot the Wannier functions in real space, we need instead to know the unk - in the code: files UNK)

Silicon, GaAs, Amorphous Silicon, Benzene 

M. Fornari, N. Marzari, M. Peressi, and A. Baldereschi, Comp. Mater. Science 20, 337 (2001)

The localisation procedure

• Long-range decay: Wannier functions corresponding to isolated valence bands decay to zero exponentially with the distance from their center

• At the global minimum (maximally-localized WFs)the Wannier functions are real (the code prints the max. absolute ratio ofimaginary and real part to check this)

• We might find a local minimum! Care is needed

• If we expect (from physical/chemical considerations)the shape and position of Wannier functions, we can give an initial guess in the form of projections on localised orbitals

Real‐Space Projectors 

Band structure interpolation

|Rni =Z

BZ

X

m

U (k)mn mk(r)e

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Maximally-localized Wannier functions

Wannier functions are defined by:

Where the Umn are chosen by the minimisation procedure (one per every k-point in the ab-initio grid, typically relatively coarse, e.g. 6x6x6)

-5

0

5

10

15

20

HWnm(k0) =

X

R

eik0·R h0n|H|Rmi

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Conversely, we can Fourier-interpolate the Hamiltonian at any k’ vector even outside the original coarse grid:

h0n|H|Rmi =X

k

e�ik·R[U †(k)H(k)U(k)]<latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit><latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit><latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit>

where the Hamiltonian matrix elements are obtained from Fourier interpolation of the initial ab-initio Hamiltonian matrix, after rotating the basisset with the unitary U matrices.

Band structure interpolation

|Rni =Z

BZ

X

m

U (k)mn mk(r)e

�ik·Rdk<latexit sha1_base64="jmzzkjSnjYP8gUzmkTcuMsdvsBc=">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</latexit><latexit sha1_base64="jmzzkjSnjYP8gUzmkTcuMsdvsBc=">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</latexit><latexit sha1_base64="jmzzkjSnjYP8gUzmkTcuMsdvsBc=">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</latexit>

Maximally-localized Wannier functions

Wannier functions are defined by:

Where the Umn are chosen by the minimisation procedure (one per every k-point in the ab-initio grid, typically relatively coarse, e.g. 6x6x6)

-5

0

5

10

15

20

HWnm(k0) =

X

R

eik0·R h0n|H|Rmi

<latexit sha1_base64="rYJAmrF23H29vL54/16pyBDtquM=">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</latexit><latexit sha1_base64="rYJAmrF23H29vL54/16pyBDtquM=">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</latexit><latexit sha1_base64="rYJAmrF23H29vL54/16pyBDtquM=">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</latexit>

Conversely, we can Fourier-interpolate the Hamiltonian at any k’ vector even outside the original coarse grid:

h0n|H|Rmi =X

k

e�ik·R[U †(k)H(k)U(k)]<latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit><latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit><latexit sha1_base64="tRag862oAKRhQhB8nOg+QSj68Ko=">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</latexit>

where the Hamiltonian matrix elements are obtained from Fourier interpolation of the initial ab-initio Hamiltonian matrix, after rotating the basisset with the unitary U matrices.

The maximal localisation tries to make sure that the matrix elements of Wannier functionsthat are far away go quickly to zero.

In this way, the Fourier interpolation is very accurate

(choosing a 6x6x6 k-grid in the ab-initio calculation corresponds to cutting to zero matrix elements beyond a

6x6x6 supercell in real space)

Disentanglement of Ahached Bands 

–  Maximally‐localized Wannier‐like func:ons for conduc:on subspace 

–  Extract differen:able manifold with op4mal smoothness 

I. Souza, N. Marzari and D. Vanderbilt, Phys. Rev. B 65, 035109 (2002)

5 d orbitals

d Bands of Copper  

s Band of Copper 

Disentanglement 

Why Minimize ΩI ? 

Decomposi:on of the Localiza:on

 Func:onal Decomposi:on of the Localiza:on

 Func:onal 

Exact Constraints on the Inner Energy

 Window 

Silicon: Bonding and An:bonding

 Orbitals 

Disentanglement with a frozen window is also useful in an insulator/semiconductor

The case of conduction bands of silicon

With two independent Wannierizations

(valence & conduction)

With a single Wannierization forvalence+conduction

Disentanglement: Conduction Bands in (5,5) SWNT